WASHINGTON (AP) — The first hurdle cleared with deceptive ease, the Senate turns to the heart of the battle over curbing gun violence next week when it considers a proposal to expand required federal background checks to gun shows and online firearms sales.

In a bipartisan 68-31 vote Thursday, senators rejected an effort by conservatives to block debate on Democrats' gun control legislation, a measure backed by President Barack Obama. Senators then formally opened debate on the bill, lawmakers' response to the mass shooting in December at an elementary school in Newtown, Conn., and the most ambitious effort to limit gun violence in nearly two decades.

Outside groups on both sides of the issue were using the weekend to try raising their visibility and pressuring lawmakers.

The National Rifle Association, a foe of the Democratic bill, was sponsoring a NASCAR race in Fort Worth, Texas, and using tweets to urge its supporters to watch it on television. Sen. Chris Murphy, D-Conn., has asked the Fox network not to broadcast the event, but it was still scheduled to be televised.

Wounded former Rep. Gabrielle Giffords, D-Ariz., and her husband, retired astronaut Mark Kelly, emailed supporters that they would lobby senators in Washington next week to broaden background checks. Obama's campaign organization, Organizing for Action, and Mayors Against Illegal Guns, financed by New York City Mayor Michael Bloomberg, were staging news conferences in at least 14 states.

Thursday's one-sided Senate vote belied what looks to be a difficult path in Congress for gun restrictions. Most Republican senators and many moderate Democrats oppose or are wary of curbs they think go too far, and the view from the GOP-run House is even cooler, where leaders say they want to first see what the Senate does.

"Nothing is going to happen quickly," Senate Majority Leader Harry Reid, D-Nev., said in a brief interview. "We're going to be on this for a while, and there's nothing wrong with that."

Asked how long debate will last, he said, "Weeks."

The bill would subject almost all gun buyers to background checks, stiffen federal laws barring illicit firearms sales and provide slightly more money for school safety measures. Background checks are aimed at preventing criminals and mentally ill people from getting weapons, and gun control advocates consider broadening the system to be the most effective step available to lawmakers.

Opponents including the National Rifle Association say the measures would infringe on the constitutional right to bear arms and inconvenience law-abiding citizens while being easy for criminals to evade. Driving that home, Sen. Mike Lee, R-Utah, read letters Thursday on the Senate floor from gun owners describing how their weapons help keep them safe.

"These are the rights we're trying to protect," he said.

But advocates of new gun curbs said after Newtown and other recent mass shootings it is time for Congress to act.

"Americans are looking to us for solutions and for action, not filibustering or sloganeering," Senate Judiciary Committee Chairman Patrick Leahy, D-Vt., said.

Excluded from the bill and facing near-certain defeat in upcoming votes were proposals to ban military-style weapons and high-capacity ammunition magazines — factors in the Newtown killings and some other recent mass shootings.

On Tuesday, senators will debate a more modest expansion of the background checks that has won bipartisan support and bolstered gun control supporters' hopes of pushing something through Congress.

The checks are now required only for sales through licensed gun dealers, while the amendment would subject all sales at gun shows and advertised forums like the Internet to the requirement, exempting noncommercial transactions such as those between relatives.

In a win for gun-rights advocates, the amendment also would make it easier to transport firearms across state lines and gradually shorten the maximum waiting time — from three days to one — for people whose gun applications are being processed by the background check system.

"All law-abiding gun owners in America should love this piece of legislation," Sen. Joe Manchin, D-W.Va., said in an interview.

As debate proceeds, some potential amendments could broaden gun rights and weaken supporters' backing for the overall bill.

One proposal is by Sens. Lindsey Graham, R-S.C., and Mark Begich, D-Alaska, who say it would improve how the background check system keeps weapons from people with certain mental problems, though critics say it would make it harder in some cases to do so. Another possible amendment would require states to recognize permits for carrying concealed weapons issued by other states.

In Thursday's vote, 50 Democrats and two Democratic-leaning independents were joined by 16 Republicans in voting to begin debate on the overall legislation.

Twenty-nine Republicans and two Democrats facing re-election next year in GOP-leaning states voted "no" — Alaska's Begich and Mark Pryor of Arkansas.